Bob is trying to decide between two cell phone plans. Plan A has no flat fee, but the user must pay $10$ cents per minute on the phone. Plan B requires a one-time fee of $\$20$, but only requires a payment of $5$ cents per minute on the phone. What is the minimum whole number of minutes Bob has to use the phone for to make Plan B the cheaper plan?
Let $x$ be the number of minutes that Bob anticipates using. Plan A costs $10x$ cents, while Plan B costs $2000 + 5x$ cents. We therefore have the inequality \begin{align*}
2000+5x &< 10x \\
\Rightarrow\qquad 2000 &< 5x \\
\Rightarrow\qquad 400 &< x.
\end{align*} The smallest integer $x$ such that $400 < x$ is $401$. Therefore, Bob must use at least $\boxed{401}$ minutes to make Plan B cheaper.